A Cosmology Independent Calibration of GRB Luminosity Relations and the Hubble Diagram

Speaker:Speaker: Liang Nan

Collaborators: Wei Ke Xiao, Yuan Liu, Shuang Nan Zhang

Tsinghua Center for Astrophysics , Tsinghua UniversityTsinghua Center for Astrophysics , Tsinghua University

National Cosmology Workshop “Constraints on Dark Energy by including GRBs”Constraints on Dark Energy by including GRBs”

Dec, 10th, 2008, IHEP

Outline• ( ) Ⅰ Introduction We present a new method to calibrate GRB relations in

a cosmology independent way.

• ( ) Ⅱ Calibration of GRB Luminosity Relations We obtain the distance modulus of a GRB at a given re

dshift from the SNe Ia data and calibrate five GRB relations without assuming a particular cosmological model.

• ( ) Ⅲ The Hubble Diagram of GRBs We construct the GRB Hubble diagram to constrain co

smological parameters.

• ( ) Summary and DiscussionⅣ

• Gamma-Ray Bursts (GRBs) are the most intense explosions observed so far.

→ GRBs are likely to occur in high redshift range.

• GRB luminosity relations are connections between measurable properties of the γ-ray emission with the luminosity or energy.

→ Recent years, several power law GRB relations have been proposed in many works.

→ Many authors have made use of GRB relations as “standard candles” at very high redshift for cosmology research.

( ) Ⅰ Introduction

see e.g. Ghirlanda, Ghisellini, & Firmani (2006), Schaefer (2007) for reviews

• The most important point related to the use of GRBs in cosmology:

---- The dependence of the cosmological model of the calibration of GRB relations.

• SN Ia cosmology: ---- adequate sample at low-redshift which can be used to

calibrate the Phillips relation essentially independent of any cosmology.

• GRB cosmology : ---- difficult to calibrate the relations using a low-z sample. ---- Calibration of GRB relations so far have been derived

by assuming a particular cosmology (e.g. the ΛCDM model).

Cosmology-dependent Calibration of GRB relations

The circularity problem

• In order to investigate cosmology, the relations of standard candles should be calibrated in a cosmological model independent way.

----Otherwise the circularity problem can not be avoided easily.

• In principle, the circularity problem can be avoided in two ways (Ghirlanda et al. 2006):

(i) A solid physical interpretation of these relations which

would fix their slope independently from cosmology. (ii) The calibration of these relations by several low reds

hift GRBs.

• Many previous works treated the circularity problem by means of statistical approaches.

→ simultaneous fit (Schaefer 2003: ) the parameters in the calibration curves and the cosmolog

y should be carried out at the same time. → Bayesian method (Firmani et al. 2005: ) → Markov Chain Monte Carlo global fitting (Li et al. 2008 ) ---- Because a particular cosmology model is required in doin

g the joint fitting, the circularity problem is not solved completely by means of statistical approaches.

Statistical approaches

• SN Ia cosmology: the distance of nearby SN Ia used to c

alibrate the Phillips relation can be obtained by measuring Cepheids in the same galaxy.

Thus Cepheids has been regard as the first order standard candle to calibrate SNe Ia as the secondly order standard candle.

• Distance of SN Ia obtained directly from observations are completely cosmological model independent.

• It is obvious that the sources at the same redshift should have the same luminosity distance for any certain cosmology.

• If distance modulus of GRBs can be obtained direct from the SN Ia data, we can calibrate the relations of GRBs in a cosmology independent way.

SNe Ia → GRBs

Cosmic distance ladder

Cepheids → SNe Ia

Extra-galactic

Cepheids

High-z GRBCosmology

Nearby SN IaStandard Candle

GalacticCepheids

Low-z SN IaCosmology

Low-z GRBStandard

Candle

z~1.7 6.6

1998: Discovery of Dark Energy

1929: Discovery of cosmic expansion

<0.1

Cosmic distance ladder: Cepheids → SNe Ia → GRBs

• There are so many SN Ia samples that we can obtain the distance modulus at any redshift in the redshift range of SN Ia directly from the Hubble diagram of SN Ia.

→ Interpolation Method

→ Iterative Method

On the basis of smoothing the noise of supernova data over redshift, a non-parametric method in a model independent manner to reconstruct the luminosity distance at any redshift in the redshift range of SNe Ia.

Methods

• If regarding the SN Ia as the first order standard candle, we can obtain the distance modulus of GRBs in the redshift range of SN Ia and calibrate the relations of GRBs in a completely cosmology independent way.

• By utilizing the relations to the GRB data at high redshift, we can use the standard Hubble diagram method to constrain the cosmological parameters.

( ) Ⅱ Calibration of GRB Luminosity Relations

• We adopt the data of 192 SNe Ia (Davis et al. 2007).

• SN1997ff (z = 1.755): the only one SN Ia point at z>1.4, therefore we initially exclude it from our SN Ia sample.

• With the GRB sample at z<1.4 obtained from SNe Ia data, GRB luminosity relations can be calibrated in a completely cosmology independent way.

Methods

We follow the iterative procedure and adopt the results from Wu & Yu, 2008

The best fitting result is obtained by minimizing

→ Interpolation Method

→ Iterative Method

(the GRB sample at z<1.4: interpolating from SNe Ia data)

(the GRB sample: iterating from SNe Ia data)

GRB Luminosity Relations Calibration

(1) : the spectral lag - luminosity relation

(2) : the variability - luminosity relation

(3) : the peak spectral energy - luminosity relation

(4) : the peak spectral energy - the collim

lag

p

p

L

V L

E L

E E

ation-corrected energy relation

(5) : the minimum rise time - luminosity relationRT L

log logy a b x

A GRB luminosity relation can be generally written in the form of

Five GRB Relations (Schaefer 2007, 69 GRBs)

(1) lag - L relation (Norris, Marani, & Bonnell 2000)

1: log ( (1 ) / 0.1lag lagL L a b z s

(Schaefer 2007, 38 GRBs)

(2) Variability - L relation (Fenimore & Ramirez-Ruiz 2000)

: log ( (1 ) / 0.02V L L a b V z

(Schaefer 2007, 51 GRBs)

(3) Epeak- L relation (Schaefer 2003; Yonetoku et al. 2004)

: log ( (1 ) / 300p pL E L a b E z keV

(Schaefer 2007, 64 GRBs)

the so-called Ghirlanda relation

(4) E γ -Epeak relation (Ghirlanda, Ghisellini & Lazzati 2004)

: log ( (1 ) / 300p pE E E a b E z keV

(Schaefer 2007, 27 GRBs)

(5) τRT - L relation (Schaefer, 2007)

τRT : the minimum rise time in the GRB light curve

1 : log ( (1 ) / 0.1RT RTL L a b z s

(Schaefer 2007, 62 GRBs)

Table 1. Calibration results (a: intercept, b: slope) for the seven GRB relations with the sample at z < 1.4 by using two methods from SNe Ia data and by assuming the LCDM.

→ Bisector: In order to avoid specifying “dependent” and “independent” variables→ OLS (ordinary least-squares)→ WLS (weighted least-squares) : taking into account the measurable uncertainties,

results in almost identical best fits.

( ) Ⅲ The Hubble Diagram of GRBs

• We use the same method used in Schaefer (2007) to obtain the best estimate μ for each GRB which is the weighted average of all available distance moduli.

• The derived distance modulus for each GRB is2

2

( / )i

i

ii

i

where the summations run from 1 to 5 over the fiverelations used in Schaefer (2007) with available data.

2 1/ 2( )i

i

with its uncertainty

SN1997ff (z = 1.755)

Black circles ： Interpolation Method

blue stars ： Iterative Method

Concordance model

→ GRB high-z “data” , obtained from the calibrated GRB “standard” candles; These data are used to fit cosmological parameters at high-z.

→ GRB low-z “data”, obtained from SN Ia data, (thus also cosmology independent). These data are used to calibrate the GRB “standard” candles.

→ SNe Ia data (Davis et al. 2007), directly from observations, cosmology independent. These data used to obain the distance moduli of GRB low-z “data”,

z = 1.4

Fig 1. Hubble Diagram of SNe Ia and GRBs

Cosmological Constrains : ΛCDM model

242

, ,22

1

,, ,

i

Nth i M obs i

Mi

CC

Chisquare statistic

(a) ： Interpolation Method

(b) ： Iterative Method

for the flat ΛCDM model

→ WLS Biesctor

Cosmological Constrains : WCDM model

242

, 0 ,20 2

1

,, ,

i

Nth i M obs i

Mi

w Cw C

(a) ： Interpolation Method

(b) ： Iterative Method

For the dark energy model with a constant equation of state

→ WLS Biesctor

( ) Summary and DiscussionⅣ

• With the basic assumption that objects at the same redshift should have the same luminosity distance, we can obtain the distance modulus of a GRB at given redshift by interpolating from the Hubble diagram of SNe Ia at z <1.4.

• There are not significant differences between the calibration results obtained by using our methods and those calibrated by assuming one particular cosmological model with the same GRB sample at z <1.4. This is not surprising, since the cosmological model is consistent with the data of SNe Ia.

• We stress again that the luminosity relations we obtained here are cosmology independent completely.

• We construct the GRB Hubble diagram and constrain cosmological parameters by the minimum χ2 method as in SN Ia cosmology.

→for the flat ΛCDM model

Interpolation Method ΩM = 0.28+0.05−0.06

Iterative Method ΩM = 0.31+0.06−0.06

→for DE model with a constant EoS

Interpolation Method w0 = −0.94+0.27−0.45

Iterative Method w0 = −0.82+0.24−0.36

----Our result suggests the the concordance model (w0 = −1, Ωm = 0.27, ΩΛ = 0.73) which is mainly derived from observations of SNe Ia at lower redshift is still consistent with the GRB data at higher redshift up to z = 6.6.

• Further examinations should be required for considering GRBs as standard candles to cosmological use.

→ It is possible that some unknown biases of SN Ia luminosity relations may propagate into the interpolation results. Nevertheless, the observable distance of SNe Ia used in our methods to calibrate the relations of GRBs are completely independent of any given cosmological model.

→ Recently, there are possible evolution effects and selection bias claimed in GRB relations.

• We emphasize that our method avoids the circularity problem more clearly than previous cosmology dependent calibration methods. Therefore our results on these GRB relations are less dependent on a prior cosmology.